A fast algorithm for spherical basis approximation

Radial basis functions appear in a wide field of applications in numerical mathematics and computer science. We present a fast algorithm for scattered data interpolation and approximation on the sphere with spherical radial basis functions of different spatial density. We discuss three settings, each leading to a special structure of the interpolation matrix allowing for an efficient implementation using discrete Fourier transforms. A numerical example is given to show the advantages of spherical radial basis functions with different spatial densities.